where \(H(z-z_{\rm N})\)  is the Heaviside function [1]} and
\(c^{\dagger }(\mathbf {r},\sigma )\)  creates a particle at point \(\mathbf {r}\)  with spin
projection \(\sigma \) . By construction, the operator (REF ) counts the
number of particles in the right fragment only, and it is easy to check that
its expectation value is \(\mathinner {\langle {\hat{X}_{\rm R}}\rangle } = \bar{X}_{\rm R}\)  as
defined in (REF ). From the definition (REF ) specialized
to the case (REF ) with the operator (REF ), we can then
introduce the projection operator \(\hat{P}^{(\rm R)}_{\rm X}(X_{\rm R})\)  that
extracts the probability that a wave function contains \(X_{\rm R}\)  particles in
the right fragment. This allows us to define the probability to find
\(X_{\rm R}\)  particles at point \(\mathbf {q}\)  in the collective space, given a
total number of particles \(X\)  in the compound nucleus
\(\mathbb {P}(X_{\rm R}| X, \mathbf {q})=\frac{\mathinner {\langle {\Phi (\mathbf {q}) | \hat{P}^{(\rm R)}_{\rm X}(X_{\rm R}) \hat{P}_{\rm X}(X) | \Phi (\mathbf {q})}\rangle }}{\mathinner {\langle {\Phi (\mathbf {q}) | \hat{P}_{\rm X}(X) | \Phi (\mathbf {q})}\rangle }}\) 
